Reading Football Rates and Starting Values

• Project a simple graph or display the table in the opening; students need notebooks or whiteboards.
• Prepare the three short football-context problems for independent work on the board or as a single handout.

Call on contrasting answers without confirming immediately. Ask, “Could the team really score 7 points every minute in a real game? Does that change what the number means in this model?” Establish that a model can simplify reality, but units and the story still determine meaning. Recall: slope is change in y divided by change in x. Today’s key move is attaching both quantities and units to the story.

Make a two-column chart: Number from representation / Meaning in the situation. Model from a table: Minutes since game started (x): 0, 5, 10, 15 Concession stand revenue (y): $0, $120, $240, $360

Then use the opening equation. “The coefficient 7 is points per minute: the predicted scoring rate. The y-intercept 14 is points when x = 0, the start of the fourth quarter.” Name the common error: students reverse the meanings because they see the intercept as the ‘first number’ or call slope simply “points.” Correct it with the question, “What happens for each 1 unit of x?” for rate, versus “What is y when x is 0?” for intercept. Compare the intercept to a game’s opening scoreboard: it is the score already on the board when the clock for this situation starts.

Display a graph labeled “Water bottles remaining at the team bench.” x-axis: minutes after halftime; y-axis: bottles remaining. The line passes through (0, 30), (10, 20), and (20, 10).

Students first write independently: slope, y-intercept, and two complete context sentences. Provide this frame for students who need it: “The rate of change is bottles per minute. This means . The y-intercept is bottles. This means when .”

Discuss using evidence from two points. Press: “Why is the slope negative?” Expected reasoning: the bottle count decreases by 10 bottles every 10 minutes, or 1 bottle per minute; it is -1 bottle per minute because bottles are being used. Confirm that -1 is not “negative bottles”; it describes a decrease. Quick CFU: show a second graph passing through (0, 8) and (4, 20), labeled “Fan-club donation total, dollars, after t weeks.” Students hold up a response showing: rate = $3 per week; intercept = $8, then explain one value to a partner. Listen for units and the x = 0 meaning; immediately restate and revise any unitless explanations.

Students solve the three focused problems, showing slope/intercept work and writing both meanings in complete sentences.

1. Table: Hours after the stadium opens: 0, 1, 2, 3. Parking fees collected: $450, $600, $750, $900.
2. A graph of “yards remaining until a running back reaches 100 yards,” with line through (0, 40), (2, 28), and (4, 16), where x is number of carries.
3. A booster club begins with $275 and earns $35 for each ticket bundle sold. Let x be bundles sold and y be total dollars. Identify and interpret rate and intercept; then explain why $35 is not the total after 35 bundles.

Circulate with a prompt rather than supplying answers: “Say the units after the number,” then “What is happening when x equals zero?” Students ready for more create a realistic football situation with a negative rate and nonzero intercept, write its equation, and explain when the model would stop making sense (for example, bottles cannot drop below zero).